Question: Simplify the following expression and state the condition under which the simplification is valid: $p = \dfrac{z^2 + 4z}{z^2 + 5z + 4}$
First factor the expressions in the numerator and denominator. $ \dfrac{z^2 + 4z}{z^2 + 5z + 4} = \dfrac{(z)(z + 4)}{(z + 1)(z + 4)} $ Notice that the term $(z + 4)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z + 4)$ gives: $p = \dfrac{z}{z + 1}$ Since we divided by $(z + 4)$, $z \neq -4$. $p = \dfrac{z}{z + 1}; \space z \neq -4$